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Green Simulation:Reusing the Output of Repeated Experiments

Mingbin Feng, Jeremy StaumDepartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 60208

[email protected], [email protected]

We introduce a new paradigm in simulation experiment design and analysis, called “green simulation,”

for the setting in which experiments are performed repeatedly with the same simulation model. Green

simulation means reusing outputs from previous experiments to answer the question currently being asked

of the simulation model. As a first method for green simulation, we propose estimators that reuse outputs

from previous experiments by weighting them with likelihood ratios, when parameters of distributions in the

simulation model differ across experiments. We analyze convergence of these green simulation estimators as

more experiments are repeated, while a stochastic process changes the parameters used in each experiment.

We find that green simulation reduces variance by more than an order of magnitude in examples involving

catastrophe bond pricing and credit risk evaluation.

Key words : likelihood ratio method; multiple importance sampling; score function method; simulation

metamodeling

1. Introduction

Consider a setting in which simulation experiments are performed repeatedly, using the same

simulation model with different values of its inputs. As we discuss in detail below, such settings

occur when a simulation model is used routinely to support a business process, and over the lifecycle

of a simulation model as it goes from development to application in repeated simulation studies. In

these settings, the standard practice is that each new simulation experiment is designed to answer

a particular question without using the output of previous simulation experiments. We advocate

a paradigm of green simulation for repeated experiments, meaning that one should reuse output

from previous experiments to answer new questions. The benefit of green simulation is greater

computational efficiency. In this article, we show that when old simulation output is reused well,

it provides greater accuracy when combined with a new simulation experiment than would be

achieved by the same new simulation experiment alone.

Green simulation entails a new perspective on management of simulation experiments. The

standard practice is to discard or ignore the output of a simulation experiment after it has delivered

the desired answer. When a new question arises, a new simulation experiment is designed to answer

it without using the output of previous simulation experiments. From this perspective, running

1

2 Feng and Staum: Green Simulation

the simulation model is a computational cost or expense. From the green simulation perspective,

running the simulation model is a computational investment that provides future benefits, because

simulation output is a valuable resource to be used in answering questions that will be asked of

the simulation model in the future.

One setting of repeated simulation experiments occurs when simulation supports a business

process. For example, in finance and insurance, simulation models support pricing and risk man-

agement decisions that are made periodically. At each period, current information, such as prices

and forecasts, is used to update the inputs to the model, and a simulation experiment is performed

to answer a question about price or risk. Similarly, in manufacturing, service, and logistics systems,

simulation models can be used routinely to provide information about expected completion times

and to support decisions about such matters as dispatching and staffing. A simulation experiment

is run whenever information is required, using inputs that describe the current state of the system.

Another setting of repeated simulation experiments occurs in the lifecycle of a simulation model

as it goes from development to application in simulation studies. First, experiments are performed

for purposes of verification and validation of the model. They may also be performed for model

calibration: to choose realistic values of unknown inputs. For these purposes, the simulation model

is run many times with different values of its inputs, to see how its outputs change with its

input, and where this behavior is reasonable and realistic. Once model development is complete,

experiments are performed for purposes such as making predictions for particular values of the

inputs, metamodeling, sensitivity analysis, and optimization. Moreover, some simulation models are

used in many simulation studies. Consequently, each model is used in several or many experiments.

Figure 1 illustrates a sequence of two repeated experiments, each with a single run. To clarify

our terminology, by “a run,” we mean one or more replications of simulation output generated

with the inputs to the model held fixed. By a simulation “experiment,” we mean a collection of

one or more runs of a simulation model, designed for the purpose of answering a specific question.

In Figure 1, the first experiment has a single run with input x1, and the second experiment has

a single run with input x2. Each run has r replications. The purpose of the nth experiment is

to estimate µ(xn), the mean output of the simulation model when the input is xn. The standard

practice is to estimate µ(x2) in the second experiment using only the single run with input x2. The

green question mark in Figure 1 indicates the question answered in this article: “How can we reuse

the simulation output from the first experiment to improve our answer in the second experiment?”

Feng and Staum: Green Simulation 3

x1

Y(1)1

· · · Y(r)1

F(Y

(1)1

)· · ·F

(Y

(r)1

)

x2

Y(1)2

· · · Y(r)2

F(Y

(1)2

)· · ·F

(Y

(r)2

)

µ(x2)

time

simulation

observed states

random vectors

outputs

performance measure

Figure 1 Setting of repeated simulation experiments.

The core idea of green simulation, reusing simulation output, has been applied to isolated exper-

iments that contain multiple runs. Many types of simulation experiments use multiple runs to learn

about the model’s response surface, the function that maps the model’s inputs to a performance

measure. In stochastic simulation, this performance measure is often the expected output of the

simulation model. There are metamodeling and sensitivity analysis experiments that run the model

at different input values to learn about how the response surface varies, globally or locally. In nested

simulation experiments, an outer-level simulation generates random values of the inputs at which

it is desired to learn the value of the response surface of an inner-level simulation. For example, in

assessing the impact of uncertainty about a simulation model’s input on the conclusions of a simu-

lation study, this model is the inner-level simulation model, and the outer-level simulation samples

values of the inputs from an appropriate distribution. In optimization via simulation experiments,

the model is run at different input values in a search for optimal input values. If simulation output

from runs at some values of the inputs can be reused in estimating the value of the response surface

at another value of the inputs, then experiment designs that involve multiple runs can be modified

to be cheaper. It is unnecessary to run many replications at every value of the inputs for which it

is desired to estimate the value of the response surface if estimates of these values can reuse output

from runs at other values of the inputs. For example, the nested simulation methods of Barton

et al. (2014) and Xie et al. (2014) use metamodeling to reuse output from a moderate number of

runs to estimate the value of the response surface for many values of the inputs, thus reducing the

required number of runs in the simulation experiment. The score function method, also known as

the likelihood ratio method (Rubinstein and Shapiro 1993, Kleijnen and Rubinstein 1996), has also


been applied to reuse output within isolated experiments for metamodeling, sensitivity analysis,

and optimization.

We adopt the likelihood ratio method as the tool for reusing output across experiments. In

applications and surveys of the likelihood ratio method that we have seen, such as Beckman and

McKay (1987), Rubinstein and Shapiro (1993), Kleijnen and Rubinstein (1996), and Glasserman

and Xu (2014), there is an isolated experiment within which each estimator reuses the output

from a single run. The exception is Maggiar et al. (2015): they construct an estimator by reusing

output from multiple runs in an isolated experiment. We reuse output from multiple runs that

come from multiple experiments. Even in the simple setting of a single run per experiment, which

we analyze in this article, reusing output from multiple experiments entails reusing output from

multiple runs. Estimation using multiple simulation runs, weighted based on likelihood ratios, was

given the name multiple importance sampling by Veach (1997). On multiple importance sampling,

see, for example, Hesterberg (1988, 1995), Owen and Zhou (2000), Veach and Guibas (1995), and

Veach (1997). Drawing on this literature on multiple importance sampling, Maggiar et al. (2015)

and we use the same two estimators, which we call the individual likelihood ratio (ILR) estimator

(Section 2.1) and the mixture likelihood ratio (MLR) estimator (Section 2.2).

The work of Maggiar et al. (2015) and our work differ in setting and findings. The key differences

are that their work focuses on an isolated experiment with a deterministic simulation model,

whereas ours focuses on repeated experiments with a stochastic simulation model. Their goal is

optimization of the response surface after smoothing by convolution with a Gaussian kernel, to

reduce the influence of numerical noise in the deterministic simulation. They apply the likelihood

ratio method to the corresponding Gaussian random variable. Their convergence theorem describes

convergence to an optimal solution within an isolated optimization experiment with an increasing

number of iterations. Our green simulation paradigm applies broadly to stochastic simulation, and

we emphasize the setting of repeated experiments. Our convergence theorems describe convergence

of estimators of values of the response surface at some or all points as the number of repeated

experiments increases. Another difference between the work of Maggiar et al. (2015) and our work

is in the findings about the ILR and MLR estimators. They report that the the choice between

ILR and MLR estimators makes “only a small difference on the performance” of their optimization

procedure when applied to a testbed of optimization problems. In Section 4, we find that the

difference between ILR and MLR estimators could be large, depending on the simulation model

and sequence of repeated experiments. We recommend the MLR estimator, which is theoretically

superior, because we find that it can work well in practice even in cases where the ILR estimator

yields poor results.


The main contribution of this article is to introduce the paradigm of green simulation for repeated

experiments and to demonstrate theoretically and experimentally that it yields significant benefits

in computational efficiency. Specifically, we investigate the case in which a simulation experiment

is run routinely (e.g., to support a business process) with updated values of inputs that serve

as parameters of distributions of random variables generated in the simulation. For this case, we

propose (Section 2), analyze (Section 3), and test (Section 4) green simulation estimators based

on the likelihood ratio method. In particular, we prove novel theorems about how these green

simulation estimators converge as the number of repeated experiments increases, while the number

of simulation replications per experiment remains constant. In this setting, the estimator based on

standard practice does not converge at all, but our green simulation estimators converge at the

canonical rate for Monte Carlo: variance inversely proportional to total computational budget. They

converge at this rate even though the total computational budget includes all previous experiments,

which are not obviously relevant to answering the question currently being asked of the simulation

model. A secondary contribution of this article is to import methods from multiple importance

sampling (where importance sampling is used for variance reduction) into the likelihood ratio

method (which reuses simulation output), and to provide evidence that there can be great practical

value to using the weighting scheme in the MLR estimator. The methods we develop in this article

are based on likelihood ratios, so they are applicable only when simulation experiments are repeated

with changes to parameters of distributions. Green simulation has broader applicability than the

specific methods proposed in this article. Some future research directions that will extend the scope

of applicability of green simulation are discussed in Section 5.

2. Green Simulation via the Likelihood Ratio Method

We develop green simulation procedures in a setting of repeated experiments with the same simula-

tion model and some changing parameters that affect the likelihood of simulated random variables.

The setting of repeated experiments, and the notation developed in this section, are illustrated

in Figure 1. Let Xn represent the parameters in the nth experiment, for example, prices observed

in the market on day n or forecasted arrival rates for period n. We treat {Xn : n= 1,2, · · · } as a

discrete-time stochastic process taking values in a general state space X . We use “state” to refer

to an element of X or a random variable taking values in X , and “current” to refer to quantities

associated with the nth experiment. Thus, Xn is the current state. We suppose that the current

state is observable at the current time step n, when the current experiment needs to be run, but

was not observable earlier.

Given the current state Xn, the current simulation experiment samples a random vector Yn

according to the conditional likelihood h (·;Xn). For example, h (y;x) could be the conditional


probability density for a stock’s price to be y in one year given that the stock’s current price is

x. As another example, h (y;x) could be the conditional probability density for the vector y of

interarrival times and service times given the arrival rate x, regarding service rate as fixed and

not included in x. We will write expectations in the form of integrals, which implicitly assumes

that Yn has a conditional probability density h (·;Xn), but this is not essential; the setting allows

for continuous, discrete, and mixed conditional distributions. We assume that h (·;x) has the same

support Y for all x∈X . We also assume that h (y;x) can be evaluated; it is not enough merely to

be able to sample according to this likelihood.

The simulation output or simulated performance of the stochastic system is F (Yn), where the

function F (·) : Y 7→ R represents the logic of the simulation model. For example, F (y) could be

the discounted payoff of a stock option if the stock’s price in one year is y, or F (y) could be the

average customer waiting time in a queue given the vector y of interarrival times and service times.

We have assumed that the stochastic process {Xn : n = 1,2, · · · } affects the simulation only

through the conditional likelihood h (·;Xn) of the random vector Yn. This assumption is necessary

to support the green simulation procedures, based on likelihood ratios, that are proposed in this

article. However, it is not necessary for green simulation in general.

In the current experiment, we wish to estimate the conditional expected performance µ(Xn)

of the stochastic system given the current state Xn; µ(Xn) can also be described as the current

expected performance. The expected performance for state x is

µ(x) =E [F (Yn) |Xn = x] =

∫

YF (y)h (y;x)dy, (1)

which is the same for all n. We assume that µ(x) is finite for all x∈X .

Standard practice in the setting of repeated experiments is to estimate µ(Xn) by the Standard

Monte Carlo (SMC) estimator

µSMCr (Xn) =

1

r

r∑

j=1

F(Y (j)n

), (2)

based on running r replications of the simulation model with the parameters set according to the

current state Xn. For simplicity in notation, we have assumed that the number r of replications

is fixed, but this is not essential. We refer to{F(Y (j)n

): j = 1, · · · , r

}as the output of the current

experiment.

The purpose of the current experiment is to estimate the current expected performance µ(Xn).

This is a random variable because the current state Xn was not observable at time step 0. Figures

of merit for an estimator, such as bias and variance, should be evaluated conditional on the current


state Xn. Clearly, µSMCr (Xn) is conditionally unbiased for µ(Xn), given Xn. The variance for state

x is

σ2(x) =Var [F (Yn) |Xn = x] =

∫

Y(F (y)−µ(x))

2h (y;x)dy, (3)

which is the same for all n. We assume that σ2(x) is finite for all x∈X . The conditional variance

of the SMC estimator, given the current state Xn, is σ2(Xn)/r. Thus, to reduce the conditional

variance of the SMC estimator, one must increase the number r of replications in the current

experiment.

Next, we propose two green simulation estimators that reuse the output of the n− 1 previous

experiments and combine it with the current (nth) experiment’s output. These green simulation

estimators are also conditionally unbiased. In Section 3, we show that, under some conditions,

the conditional variance of these green simulation estimators goes to zero as the number n of

experiments goes to infinity, even if the number r of replications per experiment is fixed.

2.1. Individual Likelihood Ratio Estimator

In the current experiment, the target distribution appears in the conditional expectation µ(Xn) =

E [F (Yn) |Xn] that we are estimating. The target distribution has the likelihood h (·;Xn). In a

previous experiment at time step k < n, the sampling distribution had likelihood h (·;Xk). To use

the output of a previous experiment in estimating µ(Xn) in a way that is conditionally unbiased

given the state history X1, . . . ,Xn, we can adjust the old output by using the likelihood ratio

between the target distribution and the sampling distribution:

E[h (Yk;Xn)

h (Yk;Xk)F (Yk) |X1, . . . ,Xn

]=

∫

Y

h (y;Xn)

h (y;Xk)h (y;Xk)F (y)dy=E [F (Yn) |Xn] .

Using the likelihood ratio method (Rubinstein and Shapiro 1993), we can combine the output from

all previous experiments and the current experiment into the following estimator of µ(x):

µILRn,r (x) =n∑

k=1

r∑

j=1

1

nr

h(Y

(j)k ;x

)

h(Y

(j)k ;Xk

)F(Y

(j)k

)(4a)

=n∑

k=1

1

n

1

r

r∑

j=1

h(Y

(j)k ;x

)

h(Y

(j)k ;Xk

)F(Y

(j)k

) . (4b)

In particular, µILRn,r (Xn) is our estimator of the current expected performance µ(Xn). However,

the use of the likelihood ratio enables us to estimate expected performance given a state x that

we never used in a sampling distribution. We refer to the green simulation estimator in (4) as

the Individual Likelihood Ratio (ILR) estimator. It contains likelihood ratios that each involve

one individual sampling distribution; this distinguishes the ILR estimator from the next green

simulation estimator we will propose.


The ILR estimator in (4b) can be seen as the average of n individual importance-sampling

estimators; the nth is the SMC estimator if state x=Xn. It is worth emphasizing the difference

between the present application of likelihood ratios and the typical application of importance

sampling. In importance sampling, the designer of the simulation experiment chooses the sampling

distribution with the aim of reducing variance. In this article, we do not address the choice of

sampling distribution. We assume that the sampling distribution in the current experiment is

determined by the current state Xn, as is standard practice in the setting of repeated experiments.

We use likelihood ratios not to reduce variance, but to enable reuse of simulation output based on

different sampling distributions.

Of course, the effect of likelihood ratios on variance needs to be considered. For any states

x,x′ ∈X , define the target-x-sample-x′ variance as

σ2x (x′) =

∫

Y

(F (y)

h (y;x)

h (y;x′)−µ(x)

)2

h (y;x′)dy. (5)

It could be more or less than the target-x-sample-x variance σ2x (x) = σ2(x) associated with

standard Monte Carlo; it could even be infinite. Based on (4b), given the state history

X1, . . . ,Xn, the conditional variance of the estimator of current expected performance, µILRn,r (Xn),

is∑n

k=1 σ2Xn

(Xk)/(n2r). If none of σ2

Xn(X1) , . . . , σ2

Xn(Xn) is too large, then the ILR estimator

has lower conditional variance than the SMC estimator. However, the ILR estimator could have

higher or infinite conditional variance. Considering that σ2Xn

(X1) , . . . , σ2Xn

(Xn) may be unequal,

one might try to construct a lower-variance estimator by using unequal weights instead of the

equal weights 1/n in (4b). Even better results are possible if we replace the equal weights 1/nr

in (4a) with unequal weights that depend on the random vector Y(j)k . This topic is addressed in

the research literature on multiple importance sampling, which leads to the next green simulation

estimator we propose.

2.2. Mixture Likelihood Ratio Estimator

In our setting of repeated experiments, we have r replications sampled from each of n distributions.

The collection of nr observations can be viewed as a stratified sample from an equally-weighted

mixture of these n distributions. The likelihood of the mixture is denoted by h (·;X1, . . . ,Xn), where

h (y;x1, . . . , xn) =1

n

n∑

k=1

h (y;xk) . (6)

Veach and Guibas (1995) advocated replacing the equal weights 1/nr in (4a) with “balance heuris-

tic” weights which, in our setting, are h(Y

(j)k ;Xk

)/nrh

(Y

(j)k ;X1, . . . ,Xn

). This leads to the mix-

ture likelihood ratio (MLR) estimator

µMLRn,r (x) =

n∑

k=1

r∑

j=1

1

nr

h(Y

(j)k ;x

)

h(Y

(j)k ;X1, . . . ,Xn

)F(Y

(j)k

). (7)


The MLR estimator is the likelihood ratio estimator that arises when we consider the pooled

outputs of all simulation experiments performed so far, {Y (j)k : j = 1, . . . , r, k= 1, . . . , n}, as a strat-

ified sample from a mixture distribution (Hesterberg 1995). It follows from this interpretation, or

immediately from the results of Veach and Guibas (1995, Section 3.2), that the MLR estimator is

conditionally unbiased for µ(x) given X1, . . . ,Xn.

The MLR estimator is superior to the ILR estimator in the sense that

Var[µMLRn,r (x)

∣∣X1, . . . ,Xn

]≤Var

[µILRn,r (x)

∣∣X1, . . . ,Xn

](8)

for any n, r, and x. This inequality follows immediately from Theorem A.2 of Martino et al. (2014).

Despite the inferiority of the ILR estimator to the MLR estimator in terms of variance, we continue

to consider both ILR and MLR estimators. One reason is that the ILR estimator is easier to analyze

theoretically (Section 3). Another is that it has lower computation cost (Section 2.3). This makes

it worth investigating, in Section 4, whether the MLR estimator has much less variance than the

LIR estimator in practical examples; we find that it does.

Finally, we consider an additional merit of the MLR estimator when used to estimate the current

expected performance µ(Xn). To obtain robustness against a poor choice of sampling distribution

in importance sampling, Hesterberg (1995) proposed to use a defensive mixture distribution, which

is a mixture of a proposed sampling distribution with the target distribution. When we estimate

µ(Xn), the target distribution is the nth distribution in the mixture, so the mixture is defensive. In

this case, the likelihood ratio h(Y

(j)k ;Xn

)/h(Y

(j)k ;X1, . . . ,Xn

)≤ n (Hesterberg 1995). It follows

that the conditional variance of µMLRn,r (Xn) given the state history X1, . . . ,Xn is at most nσ2(Xn).

2.3. Green Algorithms for Green Simulation Estimators

This section proposes and analyzes algorithms for the green simulation estimators in the setting

of repeated experiments. The algorithms are also green, in the sense that they store and reuse

likelihood evaluations as well as simulation output. Suppose that one simulation replication has

computational cost CF , one evaluation of a likelihood has computational cost Ch, and the computa-

tional cost of basic arithmetic operations such as addition, multiplication, and division is negligible

in comparison to these. We envision a situation in which CF is large, Ch is smaller but need not

be negligible, storage space is abundant, and memory access is fast. We consider a sequence of

experiments, indexed n= 1,2, · · · , of r replications each. For each experiment in the sequence, the

SMC, ILR, or MLR estimators of the current expected performance µ(Xn) are computed, and the

green simulation procedures store some information to be used in the next experiment. We analyze

the storage requirement and computation cost of the nth experiment.


For benchmarking purposes, we first consider the SMC estimator. It has zero storage requirement

in the sense that no information is stored from one experiment to the next. Its computation cost

is rCF .

Consider the ILR estimator µILRn,r (Xn) in (4a). The likelihood h(Y

(j)k ;Xk

)in the denominator

does not change as n increases. Therefore we propose to store and reuse likelihoods from one

experiment to the next in Algorithm 1. The storage requirements and non-negligible computation

costs for the nth experiment are shown on the right in Algorithm 1. The algorithm has storage

requirement 2nr and computation cost rCF +nrCh for the nth experiment. The linear growth rate

in n is reassuring; it suggests that it is affordable to reuse the outputs of many experiments in the

ILR estimator.

Algorithm 1 Green implementation of ILR estimator

1: for n= 1,2, · · · , do2: Observe Xn

3: Set µILRn,r (Xn)← 0

4: for j = 1, · · · , r do

5: Sample Y (j)n and evaluate F

(Y (j)n

). rCF computation

6: Append to output storage F(Y (j)n

). nr storage

7: Set µILRn,r (Xn)← µILRn,r (Xn) +F(Y (j)n

)

8: Calculate likelihood h(Y (j)n ;Xn

). rCh computation

9: Append to likelihood storage h(Y

(n)k ;Xn

). nr storage

10: end for

11: for k= 1, · · · , n− 1 do

12: for j = 1, · · · , r do

13: Retrieve F(Y

(j)k

)and h

(Y

(j)k ;Xk

)from storage

14: Calculate likelihood h(Y

(j)k ;Xn

). (n− 1)rCh computation

15: Set µILRn,r (Xn)← µILRn,r (Xn) +F(Y

(j)k

)h(Y

(j)k ;Xn

)/h(Y

(j)k ;Xk

)

16: end for

17: end for

18: Set µILRn,r (Xn)← µILRn,r (Xn)/nr and output

19: end for

For the MLR estimator µMLRn,r (Xn), a green algorithm is especially valuable. Inspection of (6)

and (7) suggests that the MLR estimator requires n2r likelihood evaluations: h(Y

(j)k ;x`

), for all

j = 1, . . . , r and k, ` = 1, · · · , n. A quadratic growth rate of computation cost in n could be an


obstacle for using the MLR estimator when reusing the output of many experiments. By storing and

reusing likelihoods from one experiment to the next in Algorithm 2, we avoid this quadratic growth

and achieve linear growth of the computation cost in n, as was the case for the ILR estimator.

Algorithm 2 has storage requirement 2nr and computation cost rCF + (2n− 1)rCh for the nth

experiment. This result for MLR is similar to the result for ILR, but MLR requires almost twice

as many likelihood evaluations.

Algorithm 2 Green implementation of MLR estimator

1: for n= 1,2, · · · , do2: Observe Xn

3: Set µMLRn,r (Xn)← 0

4: for j = 1, · · · , r do

5: for k= 1, . . . , n do

6: if k < n then

7: Retrieve F(Y

(j)k

)and h(Y

(j)k ) from storage

8: else

9: Sample Y (j)n and evaluate F

(Y (j)n

). rCF computation

10: Append to output storage F(Y (j)n

). nr storage

11: Set nh(Y (j)n )← 0

12: for `= 1, · · · , n− 1 do

13: Calculate likelihood h(Y (j)n ;X`

). (n− 1)rCh computation

14: Set nh(Y (j)n )← nh(Y (j)

n ) +h(Y (j)n ;X`

)

15: end for

16: end if

17: Calculate likelihood h(Y

(j)k ;Xn

). nrCh computation

18: Set nh(Y(j)k )← nh(Y

(j)k ) +h

(Y

(j)k ;Xn

)

19: Append to likelihood storage nh(Y (j)n ) . nr storage

20: Set µMLRn,r (Xn)← µMLR

n,r (Xn) +F(Y

(j)k

)h(Y

(j)k ;Xn

)/nh(Y

(j)k )

21: end for

22: end for

23: Set µMLRn,r (Xn)← µMLR

n,r (Xn)/r and output

24: end for


3. Convergence of Green Simulation Estimators

In this section, we analyze the convergence of the green simulation estimators as the number n

of experiments grows while the number r of replications per experiment is fixed. To this end, we

introduce further assumptions on the stochastic process {Xn : n = 1,2, . . .} that determines the

sampling distributions and on its relationship to the likelihood ratios between target distribution

and sampling distribution. Although not all of the assumptions in the theorems are transparent,

we show in the Appendix that they can be verified in a realistic example.

Our first assumption is that {Xn : n= 1,2, . . .} is ergodic. The intuition for making this assump-

tion is as follows. For any target state x ∈ X , we envision that it has a neighborhood such that,

for every x′ in this neighborhood, the sampling distribution associated with x′ is a good sampling

distribution for the target distribution associated with x. A good sampling distribution would be

one for which the target-x-sample-x′ variance σ2x (x′) defined in (5) is sufficiently small. An ergodic

process returns to this neighborhood infinitely often. The consequence is that, as n→∞, the

number of good samples to be used in estimating µ(x) also grows without bound.

For simplicity of exposition, we consider ergodic Markov chains with general state space. The

following summary is based on Meyn and Tweedie (2009) and Nummelin (2004), particularly on

Theorems 13.3.3 and 17.1.7 of Meyn and Tweedie (2009) and Proposition 6.3 of Nummelin (2004).

Definition 1. A Markov chain {Xn : n= 1,2, · · · } is called ergodic if it is positive Harris recur-

rent and aperiodic.

Theorem 1. If {Xn : n = 1,2, · · · } is an ergodic Markov chain that takes values in a general

state space X and has initial probability measure ν and transition kernel P , then there exists a

probability measure π on the Borel sigma-algebra B(X ) of X such that

limn→∞

∥∥∥∥∫

Xν(dx)P n(x, ·)−π

∥∥∥∥= 0. (9)

The ergodicity of a Markov chain and the existence and uniqueness of its stationary probability

measure π are equivalent. In the following theorem, the notation f ∈ L1(X ,B(X ), π) means that

the random variable f(X) has a finite expectation under π.

Theorem 2. If {Xn : n= 1,2, · · · } is an ergodic Markov chain with stationary probability mea-

sure π and f ∈L1(X ,B(X ), π), then

limn→∞

1

n

n∑

k=1

f(Xk) =

∫

Xf(x)dπ(x) (10)

Based on Theorem 2, Theorem 3 shows that the conditional variance of the green simulation

estimators evaluated at a fixed target state x, given the state history X1, . . . ,Xn, goes to zero at


the rate O(n−1). Because both green simulation estimators are unbiased, it follows that they are

consistent as n→∞. Theorem 3 also provides a result about unconditional variance, which may

be easier to interpret.

Theorem 3. Assume that {Xn : n= 1,2, · · · } is an ergodic Markov chain with stationary prob-

ability measure π. For any target state x∈X , if σ2x defined in (5) is in L1(X ,B(X ), π), then

limn→∞

nrVar[µMLRn,r (x)

∣∣X1, . . . ,Xn

]≤ lim

n→∞nrVar

[µILRn,r (x)

∣∣X1, . . . ,Xn

]

=

∫

Xσ2x(x′)dπ(x′), a.s. (11)

If, furthermore, the target-x-sample-Xn variance process {σ2x (Xn) , n= 1,2 · · · } is uniformly inte-

grable, then

limn→∞

nrVar[µMLRn,r (x)

]≤ lim

n→∞nrVar

[µILRn,r (x)

]=

∫

Xσ2x(x′)dπ(x′). (12)

Proof. We derive the results for the ILR estimator; the results for the MLR estimator follow

due to (8). By the conditional independence of the random vectors {Y (j)k : j = 1, . . . , r, k= 1, · · · , n}

given X1, . . . ,Xn, we have

Var[µILRn,r (x)

∣∣X1, . . . ,Xn

]=

1

n2r2

n∑

k=1

r∑

j=1

Var

F

(Y

(j)k

) h(Y

(j)k ;x

)

h(Y

(j)k ;Xk

)

=

1

n2r

n∑

k=1

σ2x (Xk) . (13)

Therefore, by Theorem 2, we obtain (11):

limn→∞

nrVar[µILRn,r (x)

∣∣X1, . . . ,Xn

]= lim

n→∞

1

n

n∑

k=1

σ2x (Xk) =

∫

Xσ2x(x′)dπ(x′), a.s.

To establish (12), consider that

Var[µILRn,r (x)

]= E

[Var

[µILRn,r (x)

∣∣X1, . . . ,Xn

]]+Var

[E[µILRn,r (x)

∣∣X1, . . . ,Xn

]]

= E[Var

[µILRn,r (x)

∣∣X1, . . . ,Xn

]]+Var [µ(x)]

= E

[1

n2r

n∑

k=1

σ2x (Xk)

],

using (13). Therefore

limnr→∞

nrVar[µILRn,r (x)

]= lim

n→∞E

[1

n

n∑

k=1

σ2x (Xk)

]=E

[limn→∞

1

n

n∑

k=1

σ2x (Xk)

]=

∫

Xσ2x(x′)dπ(x′),

where the exchange of limit and expectation holds by uniform integrability, and Theorem 2 justifies

the last step.


Theorem 3 shows the asymptotic superiority of the green simulation estimators to standard

Monte Carlo (SMC), as the number n of repeated experiments increases. Recall, from the discussion

of (3), that the conditional variance of the SMC estimator for the current expected performance

µ(Xn), given the state history, is σ2(Xn)/r. This does not converge to zero as n→∞, assuming

that sampling variances are positive. Yet the green simulation estimators converge to the true value

µ(x) as n→∞. This means that we can obtain arbitarily high accuracy without increasing the

budget r per experiment, merely by reusing output from experiments that are repeated at each time

step with budget r. Under the assumptions of Theorem 3, reusing old simulation output is highly

effective in the sense that the variance of the green simulation estimators converges as O((nr)−1),

which is the standard rate of convergence for Monte Carlo in terms of the computational budget

nr expended on all experiments that were ever run in the past.

Theorem 3 is about convergence of green simulation estimators for any fixed target state x

that satisfies certain conditions. Theorem 4 provides the conclusion that the convergence of the

ILR estimator is uniform across all states x ∈ X . That is, after a sufficiently large number of

experiments, the function µILR has small error as an estimator of the function µ that maps a state

x to the expected performance µ(x). As a corollary, it follows that the ILR estimator of the current

expected performance, µ(Xn), eventually has small error. Theorem 4 is similar to Theorem 1 of

Staum et al. (2015), but it simplifies one of the assumptions of the latter theorem by relying on

the ergodicity of {Xn : n= 1,2, · · · }.

Theorem 4. Assume that {Xn : n= 1,2, · · · } is an ergodic Markov chain with stationary prob-

ability measure π, and the following assumptions hold:

A1. The state space X is a subset of a Euclidean space. The function µ :X 7→R is bounded.

A2. The support Y is a subset of a Euclidean space. For any x∈X , the function F (·)h (·;x) is

measurable. For any y ∈Y, the function h (y; ·) is bounded on X .

A3. There exist non-negative constants a and b such that, for all x,x′ ∈X and all y ∈Y,

|F (y) | h (y;x)

h (y;x′)≤ a exp (b‖y‖2) .

A4. Let M(x, t) =∫

exp(t‖y‖2)h (y;x)dy. There exists t > b, where b is as in A3, such that

M(·, t) ∈ L1(X ,B(X ), π) and that the sequence of random variables {M(Xn, t) : n = 1,2, · · · } is

uniformly integrable.

Then limn→∞

supx∈X|µILRn,r (x)−µ(x)|= 0, almost surely.

Proof. The conclusion of this theorem is the same as that of Theorem 1 of Staum et al. (2015),

so it suffices to verify the five assumptions of that theorem. Assumptions 1, 3, and 4 of Staum

et al. (2015) hold as direct consequences of A1, A2 and A3. Their Assumption 2 holds because we


have assumed that the sampling distribution h (·;x) has a common support Y for all states x∈X .

It remains to verify Assumption 5 of Staum et al. (2015) using A4 and ergodicity.

By ergodicity, the distribution ofM(Xn, t) converges to a limiting distribution as n→∞. Because

M(·, t)∈L1(X ,B(X ), π), this limiting distribution of M(Xn, t) has a finite mean; call it m. Uniform

integrability of {M(Xn, t) : n= 1,2, · · · } implies that limn→∞

E [M(Xn, t)] = E[

limn→∞

M(Xn, t)]

=m.

Therefore there exists a finite c such that E [M(Xn, t)]≤ c for all n= 1,2, · · · . Let p= t/b > 1. We

now have∞∑

i=1

i−pE [M(Xn, pb)]≤ c∞∑

i=1

i−p <∞,

which verifies Assumption 5 of Staum et al. (2015).

Corollary 1. Under the assumptions of Theorem 4, limn→∞ |µILRn,r (Xn)− µ(Xn)| = 0 almost

surely.

Proof. From Theorem 4, it follows that for every ε > 0, there exists a random variable Nε such

that, almost surely, for all x∈X and n>Nε, we have |µILRn,r (x)−µ(x)|< ε. Therefore, for all n>Nε,

we have |µILRn,r (Xn)−µ(Xn)|< ε.

4. Numerical Examples

In this section, we use two numerical examples to illustrate green simulation, demonstrate its

value, and compare our two green simulation estimators with each other and with standard Monte

Carlo. First is a reinsurance example of pricing catastrophe bonds. For this example, we verify the

conditions of Theorem 3 and Theorem 4 in the Appendix, which shows that the conditions are

applicable to a realistic example. The experiment results conform to the theoretical predictions

and show that the MLR estimator is superior to the ILR estimator, which is superior to the

SMC estimator. The second example involves measuring the credit risk of a loan portfolio. In this

example, the conditions of the theorems do not hold, but the experiment results show that green

simulation can still deliver valuable results in such a situation. Although the ILR estimator is not

reliable in this example, the MLR estimator is superior to the SMC estimator.

4.1. Catastrophe Bond Pricing with Compound Losses

A catastrophe bond (“CAT bond”) is an important reinsurance contract that helps insurance

companies to hedge against losses from catastrophic events (Munich Re Geo Risks Research 2015).

This example is relevant beyond insurance; for example, senior tranches of structured financial

instruments are essentially economic catastrophe bonds, because they suffer credit losses only

in the event of an economic catastrophe (Coval et al. 2009). Simulation of CAT bonds can be

computationally intensive, because it involves fairly rare events in a complex geophysical models.

Specifically, the example is of a hurricane CAT bond.


In practice, reinsurance contracts are subject to periodic renewals. This is the source of the

repeated experiments: the same CAT bond is priced every period, using the same hurricane sim-

ulation model, with parameters Xn updated to reflect the current climatological forecast. In this

example, the period is semi-annual. The state process {Xn : n= 1,2, · · · } is modeled by an ergodic

Markov chain. Given the state Xn, we can simulate hurricanes that take place during the lifetime

of the CAT bond and the resulting total insured loss Yn underlying the CAT bond. Finally, we

compute the payoff of the CAT bond per dollar invested,

F (Yn) = 1{Yn≤K}+ p1{Yn>K},

where 1{·} is the indicator function, K is the trigger level, and p∈ [0,1) is the fraction of face value

that is received if insured losses exceed the trigger level. The fair price of this CAT bond can be

obtained in terms of the expected payoff µ(Xn) = E [F (Yn) |Xn] . We consider a hurricane CAT

bond with lifetime 10 years, trigger level K = 25 million dollars, and recovery fraction p= 0.5.

In this example, we use a simplified version of the model of Dassios and Jang (2003). The insured

loss is modeled as a compound random variable: Yn =Mn∑i=1

Zin, where Mn denotes the number of

claims and Zin denotes the ith claim size. In this model, Z(i)n , i= 1,2 · · · are i.i.d. and independent of

Mn. This is a popular loss model due to its flexibility and suitability for many practical applications

(Klugman et al. 2012), yet it can provide mathematical tractability. Let the probability mass

function of Mn be p(m;λn) and the probability density of Zin be f(z;θn), where λn and θn are

parameters determined by the state Xn. Specifically, we take Mn to be Poisson with mean λn and

Zin to be exponential with mean θn. In this model, the expected payoff

µ(Xn) =E [F (Yn) |Xn] =E[p+ (1− p)1{Yn≤K}

]= p+

∞∑

m=1

p(m;λn)F (K;θn,m) (14)

where p(m;λ) is the Poisson probability mass function with mean λ and F (K;θ,m) is the Gamma

cumulative distribution function with scale parameter θ and shape parameter m. From 1981-2010,

the average number of major hurricanes was 2.7 per decade and the average cost per hurricane

was about $5,000 million (Blake and Gibney 2011). Therefore, we set up a stochastic model of the

states {Xn : n= 1,2, · · · } and a transformation (λ, θ) = ψ(x) so that λn is usually around 2.7 and

θn is usually around 5 (measured in thousands of millions of dollars).

In this example, the ergodic Markov chain driving the parameters of the loss model is a stationary

AR(1) process with state space X =R2, given by

Xn = µ∞+ϕXn−1 + εn,


where {εn : n= 1,2, · · · } is an i.i.d. sequence of bivariate normal random vectors with mean zero

and variance diag(σ2ε), and the parameters are

µ∞ =

[00

], ϕ=

[0.60.5

], and σ2

ε =

[0.82

0.52

].

The state space X = R2 is inappropriate for parameters that must be non-negative because they

represent an expected number of hurricanes and an expected loss per hurricane. We introduce a

transformation ψ : R2 7→ (λ, λ)× (θ, θ) so that the parameters (λ, θ) = ψ(x) lie between plausible

lower and upper bounds. The transformation is sigmoidal and maps x= [x1, x2] to

(λ, θ) =ψ(x) =[λ+ λ−λ

1+e−x1 , θ+ θ−θ1+e−x2

].

In particular, we took (λ, λ) = (2,4) as the range for expected number of hurricanes per decade and

(θ, θ) = (4,6) as the range for expected loss per hurricane. To give a picture of the variability of the

parameters for repeated experiments in this example, Figure 2 shows histograms of the parameters

λ and θ resulting from sampling from the stationary distribution of the AR(1) process.

2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Normalized histogram of λ

4 4.5 5 5.5 60

0.5

1

1.5Normalized histogram of θ

Figure 2 Histograms of parameters sampled based on the stationary distribution of the AR(1) process.

To clarify the model, Algorithm 3 shows how the standard Monte Carlo simulation works.

To investigate the effectiveness of green simulation, we performed a sequence of 100 repeated

simulation experiments (i.e., n = 1,2, · · · ,100) with r = 100 replications each. Using the same

sample path {Xn : n = 1,2, · · · ,100} and the same simulation output {Y (j)n : n = 1,2, · · · ,100, j =

1,2, · · · ,100}, we evaluated the SMC, ILR, and MLR estimators at each period n= 1,2, · · · ,100,

in each of three states: the current state Xn, the central state xmi = (0,0) corresponding to the

moderate parameters λ= 3 and θ= 5, and an extreme state xhi = (∞,∞) corresponding to extreme

parameters λ = 4 and θ = 6. (Strictly speaking, the extreme state xhi does not belong to the


Algorithm 3 Standard Monte Carlo Simulation for Catastrophe Bond Pricing

1: Sample state X0 from the stationary distribution of the AR(1) process

2: for n= 1,2, · · · , do3: Sample state Xn from AR(1) process, conditional on Xn−1

4: Set parameters (λn, θn)←ψ(xn)

5: Set µSMCn,r (Xn)← 0

6: for j = 1, · · · , r do

7: Sample number of hurricanes M (j)n ∼ p(·;λn)

8: for i= 1, · · · ,M (j)n do

9: Sample loss for ith hurricane Z(i,j)n ∼ f(·;θn)

10: end for

11: Set Y (j)n ←

Mn∑i=1

Zi,jn

12: if Y (j)n ≤K then

13: Set F(Y (j)n

)← 1

14: else

15: Set F(Y (j)n

)← p

16: end if

17: Set µSMCxn

(Xn)← µSMCxn

(Xn) +F(Y (j)n

)

18: end for

19: Set µSMCn,r (Xn)← µSMC


20: end for

state space X = R2, but the parameter vector (λ, θ) = (4,6) is a limit point of the range of the

transformation ψ that maps states to parameters.)

For the purpose of accurately estimating the unconditional variances of the all these estimators,

we performed such a sequence of experiments 10,000 times. These 10,000 macro-replications of the

sequence of experiments have independent sample paths and simulation output. The estimated

variance of a fixed-state estimator µ(x) of µ(x) was10,000∑k=1

(µ(k)(x)− µ(x))2/10,000, where µ(k)(x)

is the value of the estimator on the kth macro-replication. Likewise, the estimated variance of

a current-state estimator µ(Xn) of µ(Xn) was10,000∑k=1

(µ(k)(X(k)n )− µ(X(k)

n ))2/10,000. Due to using

10,000 macro-replications, the standard errors of these estimated variances are less than 1% of the

corresponding estimated variance.

Figure 3 is a log-log plot of the variances of the SMC, ILR, and MLR estimators for two fixed

states, xmi and xhi, for each experiment n = 1,2, · · · ,100. The horizontal solid blue line is the

SMC variance, the dotted red line is the ILR variance, and the dashed purple line is the MLR


variance. The fixed-state SMC estimators µSMCn,r (xmi) and µSMC

n,r (xhi) were generated by sampling

according to h (·;xmi) and h (·;xmi), respectively, in experiments distinct from the experiments

described in the preceding paragraph. The SMC variance forms a horizontal line because, for each

experiment n, µSMCn,r (xmi) and µSMC

n,r (xhi) use a fixed number r of replications drawn from the

sampling distribution associated with the fixed state xmi or xhi. In addition, a black solid line with

slope −1 and intercept equal to the SMC variance is plotted for reference. This line shows the

variance of an SMC estimator with nr replications, which is the cumulative number of replications

simulated up through the nth experiment. We compare the lines for green simulation estimators

against the solid lines. When they go below the horizontal line, they have lower variance than a

SMC simulation with r replications, the budget for a single experiment. When they go near the

line with slope −1, they have variance nearly as low as a SMC simulation with nr replications, the

cumulative budget for all experiments so far. That is a major success for green simulation, because

it shows that reusing old simulation output is nearly as effective as generating new simulation

output in the current experiment.

Number of periods, n100 101 102

10-6

10-5

10-4

10-3

10-2

estimated central state (xmi) variance

SMC

ILR

MLR

Slope −1

Number of periods, n100 101 102

10-6

10-5

10-4

10-3

10-2

estimated extreme state (xhi) variance

SMC

ILR

MLR

Slope −1

Figure 3 Log-log plots of estimated variances of fixed-state estimators.

We see from Figure 3 that the MLR estimator has lower variance than the ILR estimator, which

is consistent with (8). In this example, the gap between them grows to be substantial as the number

n of repeated experiments increases: for n = 100, the ratio between ILR and MLR variances is

about 1.7 for both xmi and xhi. Initially, for very small n, the green simulation estimators have

higher variances than an SMC estimator based on a simulation in the fixed state xmi or xhi. The

cause is the probability that none of the states visited so far, X1,X2, · · · ,Xn, were near xmi or xhi.


This event is more likely for the extreme state xhi than for the center state xmi, which is why the

higher variances persist longer for the extreme state (for the first 5 experiments) than for the center

state (only for the first experiment). The variances of the green simulation soon become lower

than those of SMC and continue to decrease as the outputs of more experiments are reused. After

100 experiments, the MLR estimator’s variance is over 45 times smaller than the SMC estimator’s

variance for xhi and over 95 times smaller for xmi. In other words, by using 10,000 total replications

simulated in 100 simulation experiments, with simulation based on parameters corresponding to

X1,X2, · · · ,X100 and not xhi, the MLR estimator achieves higher accuracy in estimating µ(xhi) than

standard Monte Carlo with 9,500 replications simulated based on parameters corresponding to xhi.

Comparing to the black solid reference line, we see that the green simulation estimators’ variances

eventually decrease approximately at a rate of n−1, as discussed in connection with Theorem 3.

Next we consider the variance of the SMC, ILR, and MLR estimators for the current-state

expected performance µ(Xn). Figure 4 shows their variances. For the first experiment (n= 1), there

is no stored simulation output from a previous experiment, so all three estimators are the same.

In this example, the green simulation estimators’ variances decrease from the beginning and are

always less than the SMC variance. Otherwise, the performance of the green simulation estimators

is similar to what was seen when the state was fixed. After 100 experiments, the MLR estimator’s

variance is over 61 times smaller than the SMC estimator’s variance.

4.2. Credit Risk Management

In simulation for financial risk management, experiments using the same simulation model are

performed periodically, as often as daily. Information observed in the markets is used to update

parameters that affect risk, and the simulation model is run again with new parameters. In this

example, the risk management simulation measures the credit risk exposure of a portfolio containing

loans to companies with listed equity. The asset values of these debtor companies are observable

and serve as parameters in the risk model: the lower the asset value of a debtor company, the more

likely it is to default in the future. Thus, in our setting of repeated experiments, the current state

Xn contains the current asset values of all debtor companies.

In this example, we work with a structural model of default based on the influential work of

Merton (1974); for an exposition, see McNeil et al. (2005), for example. At any period n, the asset

value of a company equals the sum of its equity and debt values. The equity value can be observed

in the stock market and the debt value can be observed from public records, so the asset value can

be computed. For simplicity of exposition, we assume that debt remains constant. In this model,

the asset value follows geometric Brownian motion, and a company defaults when its asset value

falls below its debt. Because geometric Brownian motion is not an ergodic process, Theorems 3


Number of periods, n

100 101 10210-6

10-5

10-4

10-3

SMC

ILR

MLR

Slope −1

Figure 4 Log-log plot of estimated variances for current-state estimators.

and 4 do not apply. As n increases, the current state Xn tends to drift further away from an earlier

state, such as X1. Therefore, intuition suggests that the benefit of reusing old simulation output

would diminish over time. We consider this example to show that green simulation nonetheless

delivers valuable results.

We consider a loan portfolio whose composition remains constant over time. Many loan portfolios

are dynamic: as outstanding loans are being repaid, new loans are being initiated. Despite the

dynamic nature of such portfolios, there are lending businesses in which portfolios retain a similar

composition over time. For example, in the business of accounts receivable, customers may place

regular, periodic orders of the same size, each resulting in payment due in 90 days. An investment

fund may target a loan portfolio with fixed characteristics such as maturity and portfolio weights

on different types of loans.

In our simplified example, we consider risk management of the value at time horizon t = 0.5

years of a portfolio in which there are two loans, both having maturity T = 5 years. Simulation

experiments are repeated weekly, i.e., with a period of ∆t = 1/52 years. In the nth experiment,

the quantity µ(Xn) being estimated is the conditional probability, given the current asset values

Xn, that the cost of defaults and anticipated defaults after t years will exceed a threshold κ= 6.

The random vector Yn that we simulate in the nth experiment is the asset values St = [St,1, St,2] at

time t, given the current asset values S0 = [S0,1, S0,2] =Xn. For each company i= 1,2, the marginal


distribution of the asset return St,i/S0,i is lognormal, determined by the drift ηi and volatility ςi

of the geometric Brownian motion for asset value. Specifically, η1 = 15%, η2 = 10%, ς1 = 30%, and

ς2 = 20%. The joint distribution of the asset returns St,1/S0,1 and St,2/S0,2 is specified by a Student

t copula with 3 degrees of freedom and correlation 0.5. The initial asset values of the two companies

are X0,1 = 100 and X0,2 = 90, and their debt is D1 = D2 = 85. The loss if company i defaults is

denoted ai, and a1 = 5 and a2 = 4. The discount rate is r= 5%. The cost of a default or anticipated

default by company i, as of time t, is

Li = ì(St,i) =

{ai if St,i <Di

aie−r(T−t)Φ

(ln(St,i/Di)+(r−ς2i /2)(T−t)

ςi√T−t

)if St,i ≥Di

.

The first line of the formula represents the loss if company i defaults at time t. The second line of

the formula represents a risk-neutral conditional expectation, as of time t, of the discounted loss

at time T if company i defaults then. The portfolio’s loss is L1 + L2, the sum of losses over the

debtor companies. The simulation output F (Yn) is 1 if L1 + L2 > κ, and 0 otherwise. To clarify

the model, Algorithm 4 shows how the standard Monte Carlo simulation works.

Algorithm 4 Standard Monte Carlo Simulation for Credit Risk Management

1: for n= 1,2, · · · , do2: Sample state Xn from bivariate lognormal distribution with Student t copula, based on

time increment ∆t, conditional on Xn−1

3: Set µSMCn,r (Xn)← 0

4: for j = 1, · · · , r do

5: Sample state Y (j)n from bivariate lognormal distribution with Student t copula, based

on time increment t, conditional on Xn

6: Set L(j)n ← ì(Y

(j)n,1 ) + ì(Y

(j)n,2 )

7: if L(j)n ≤ κ then

8: Set F(Y (j)n

)← 1

9: else

10: Set F(Y (j)n

)← 0

11: end if

12: Set µSMCxn

(Xn)← µSMCxn

(Xn) +F(Y (j)n

)

13: end for

14: Set µSMCn,r (Xn)← µSMC


15: end for

To investigate the effectiveness of green simulation, we performed a sequence of 52 simulation

experiments repeated weekly (i.e., n= 1,2, · · · ,52) with r= 1,000 replications each. Using the same


sample path {Xn : n = 1,2, · · · ,100} and the same simulation output {Y (j)n : n = 1,2, · · · ,100, j =

1,2, · · · ,100}, we evaluated the SMC, ILR, and MLR estimators at each period n= 1,2, · · · ,100,

in the current state Xn. For the purpose of accurately estimating the unconditional variances of

the all these estimators, we performed such a sequence of experiments 10,000 times. These 10,000

macro-replications of the sequence of experiments have independent sample paths and simulation

output. The estimated variance of a current-state estimator µ(Xn) of µ(Xn) was10,000∑k=1

(µ(k)(X(k)n )−

µ(X(k)n ))2/10,000. These estimated variances appear in a log-log plot in Figure 5, along with vertical

error bars representing their 95% approximate-normal confidence intervals.

Number of periods, n100 101

10-6

10-5

10-4

10-3

SMC

ILR

MLR

Figure 5 Log-log plot with error bars for estimated variances for current state estimators.

Figure 5 shows behavior for the MLR estimator similar to what was seen for the previous

example in Figure 4. The MLR estimator’s variance is less than the SMC estimator’s variance,

and it decreases as the number n of repeated experiments increases. By n= 52, it is over 17 times

smaller than the SMC estimator’s variance. However, in Figure 5, we see effects of the non-ergodic

nature of the state process {Xn : n= 1,2, · · · }. Due to the positive drifts of the asset prices, debtor

companies tend to become less likely to default, so the SMC variance decreases slightly as the

number of periods n increases, instead of forming a straight line as in Figure 4. The dramatic effect

is on the behavior of the ILR estimator. Its variance decreases over the first 6 experiments, but

after about 20 experiments, its variance increases again. Eventually, its variance exceeds the SMC

estimator’s variance. Apparently, the difference between sampling distributions for state X1 and


X52 is likely to become so large that the use of likelihood ratios in the ILR estimator (4) inflates

variance. The failure of the ILR estimator and the success of the MLR estimator in this example

demonstrate the practical importance of the MLR estimator. The evident disutility of some of the

old simulation output in this example also raises a future research question in green simulation:

how to determine which old simulation output is worthwhile to reuse in estimating the expected

performance in the current state.

5. Conclusions and Future Research

In Section 3, we established theorems about the convergence of green simulation estimators as

the number of repeated experiments increases. We tested their practical performance for small

and moderate numbers of repeated experiments in two examples in Section 4. In the example

of Section 4.1, the conditions of the theorems held. Our ILR and MLR estimators were both

successful in significantly reducing variance compared to standard Monte Carlo, but MLR was

somewhat better. In the example of Section 4.2, the conditions of the theorems did not hold.

The MLR estimator was successful in significantly reducing variance, whereas the ILR estimator

had problems. Consequently, we recommend the MLR estimator for doing green simulation in

the setting analyzed in this article: where simulation experiments are repeated with changes to

parameters of distributions of random variables generated in the simulation. The variance reduction

achieved by green simulation depends on several aspects of the particular example: the stochastic

process that describes changing parameters, the particular distributions whose parameters change,

and the number of repeated experiments. Our theorems show that, when their conditions hold,

green simulation eventually becomes greatly superior to standard Monte Carlo. Our experiment

results suggest that green simulation is extremely promising: in the only two examples that we

investigated, the MLR estimator achieved variance lower than standard Monte Carlo by factors of

17 and 61 after a moderate number of repeated experiments.

Because green simulation is a new paradigm, there are several good directions for future research.

Here we call attention to a few that are most relevant to this article.

Some future research topics are relevant to the specific methods proposed in this article. We found

the MLR estimator to be satisfactory for our purposes. However, further enhancements have been

considered in the literature on importance sampling. For example, Hesterberg (1995) investigated

different ways to normalize weights, and Owen and Zhou (2000) proposed to use likelihoods that

appear in the MLR estimator as control variates. At the end of Section 4.2, the experiment results

raised the question of which old simulation output to reuse in green simulation. This question is

worthy of investigation in connection with the methods proposed in this article and also with other

methods. In general, there are two possible drawbacks to using all of the old simulation output.


One is that some of the old simulation output makes an estimator worse if it is reused than if it

is not reused (as seen in Section 4.2). The other is that if the amount of old simulation output is

extremely large, reusing more of it generates diminishing returns in terms of improved estimator

quality compared to the computational cost of reusing it. The theoretical and practical benefit of

various green simulation methods could be enhanced by good rules for selecting the subset of old

simulation output to reuse.

Various green simulation methods are possible. Besides those proposed in this article, others

should be investigated because they would be more broadly applicable or more effective in some

cases. In this article, we used the likelihood ratio method for green simulation. The likelihood

ratio method is directly applicable only when simulation experiments are repeated with changes to

parameters of distributions of random variables generated in the simulation, not when a changing

parameter affects something other than a distribution, e.g., numbers of servers or sizes of buffers in

a simulation of a queuing system. Even if these methods are applicable, they would not be highly

effective if it is unlikely to visit a state that is sufficiently similar to a previously visited state, where

similarity is measured according to (5). Rubinstein and Shapiro (1993) address the likelihood ratio

method’s effectiveness and extensions of its applicability. More broadly applicable green simulation

methods can be designed based on metamodeling. We have obtained encouraging initial results

based on stochastic kriging (Ankenman et al. 2010). Green simulation based on Database Monte

Carlo, used as in Rosenbaum and Staum (2015), is also more broadly applicable.

We focused on showing that when old simulation output is reused well, it provides greater

accuracy when combined with a new simulation experiment than would be achieved by the same

new simulation alone. We analyzed how the accuracy of an answer to the current question improves

as the number of repeated experiments increases. However, it might be possible to answer the

current question sufficiently accurately with no further experiment. If a new simulation experiment

was indeed required, one could design it in light of the current question and the currently available

information. This leads to future research in experiment design not from a blank slate. Also, take

consideration of possible future questions when designing the current one, in light of knowledge of

the state process.

Appendix. Verifying the Conditions of Theorem 3 and Theorem 4 for

the Catastrophe Bond Pricing Example

In this appendix, we verify the conditions of Theorem 3 and Theorem 4 for the catastrophe bond (CAT

bond) pricing example. In this example, the underlying state process {Xn : n = 1,2, · · · } is AR(1), so it is

ergodic. Recall that the loss in the CAT bond pricing example is given by Yn =Mn∑i=1

Zin. Conditional on Xn = x,

and denoting φ(x) = (λ, θ), the number Mn of claims is Poisson distributed with mean λ and independent of


{Zin : i= 1, · · · ,Mn}, which are independent random variables, exponentially distributed with mean θ. The

conditional distribution of Yn given Xn places probability mass h(0;x) = e−λ on y = 0 and has probability

density

h(y;x) =

∞∑

m=1

λm

m!e−λ

ym−1e−y/θ

Γ(m)θm=

√λ

θye−λ−y/θI1

(2

√λy

θ

)

for y > 0, where Γ is the Gamma function and I1 is the modified Bessel function of the first kind of order 1.

We will first establish two lemmas that are useful for verifying the conditions of Theorems 3 and 4 for the

CAT bond pricing example. Define the domain K = (0,∞)× (0,∞)×R and the functions A : K 7→ R and

a :K× (0,∞) 7→R by

A(k) =

∫ ∞

0

a(k, y)dy, where a(k, y) =e−k1y+k3

√y

√y

I1(k2

√y)≥ 0. (15)

Lemma 1. The function A is continuous on K.

Proof. For any k, k′ ∈K, we have

|A(k)−A(k′)| ≤∫∞Na(k, y)dy+

∫∞Na(k′, y)dy+

∫ N0|a(k, y)− a(k′, y)|dy.

For any k ∈K and ε > 0, we will show that there exist N > 0 and δ > 0 such that the right side is bounded

above by ε if ‖k− k′‖2 < δ. First, we will show that, for any k ∈K and ε > 0, there exists N1 > 0 such that∫∞N1a(k, y)dy ≤ ε/3. Applying the same argument to k′ ∈ K, there exists N2 such that

∫∞N2a(k′, y)dy ≤ ε/3.

We then let N = max{N1,N2}. Finally, we show that for any k ∈K, ε > 0, and N > 0, there exists some δ > 0

such that∫ N

0|a(k, y)− a(k′, y)|dy≤ ε/3 if ‖k− k′‖2 ≤ δ.

First, it is proved by Luke (1972) that Γ(ν+ 1)(2/y)νIν(y)< cosh(y) for y > 0 and ν >−1/2. Taking ν = 1

in this inequality, and observing that cosh(y)< ey, we have I1(y)< (y/2)ey. Let k1 = k1/2> 0. Then

a(k, y) =e−k1y+k3

√y

√y

I1(k2

√y)<

e−k1y+k3√y

√y

(k2√y

2ek2√y

)≤ k2e

(k2+k3)2/(4k1)

2e−k1y =:Ce−k1y,

where the second inequality holds because (k2 + k3)√y− k1y ≤ (k2 + k3)2/(4k1), and the constant C > 0 is

defined for ease of notation. Therefore∫∞Na(K,y)dy≤C

∫∞Ne−k1ydy=C(e−Nk1/k1). Take

N1 =− ln[εk1/3C]

k1

=−2 ln[(εk1)/(6C)]

k1

.

Then∫∞N1a(K,y)dy≤ ε/3.

The function I1 is a solution of Bessel’s differential equation, so it is continuous on (0,∞). Consequently, the

function a :K× (0,∞) 7→R defined as a(k, y) := e−k1y+k3√yI1(k2

√y) = a(k, y)

√y is continuous on K× [0,∞).

Choose any δ0 > 0 and define the compact neighborhood Nk(δ0) := {k′ ∈K : ‖k− k′‖2 ≤ δ0}. The function a

is continuous on the compact set Nk(δ0)× [0,N ]. Therefore, it is uniformly continuous on Nk(δ0)× [0,N ].

Consequently, there exists δ ∈ (0, δ0] such that, for all (k, y), (k′, y′) ∈ Nk(δ0)× [0,N ] that satisfy ‖(k, y)−(k′, y′)‖2 ≤ δ, we have |a(k, y)− a(k′, y′)| ≤ ε/(6

√N). Therefore, for any k′ ∈ K such that ‖k− k′‖2 ≤ δ, we

have ∫ N

0

|a(k, y)− a(k′, y)|dy=

∫ N

0

1√y|a(k, y)− a(k′, y)|dy≤ ε

6√N

∫ N

0

1√ydy=

ε

3.


Define the domain KB = (0,∞)× (0,∞)× (0,∞) and the functions B :KB 7→ R and b :KB × (0,∞) 7→ R

by

B(k) =

∫ ∞

0

b(k, y)dy, where b(k, y) =e−k1y√y

[I1(k2√y)]2

I1(k3√y)≥ 0.

Lemma 2. If K ⊂K is compact, then sup{A(K)|K ∈ K}<∞. If KB ⊂KB is compact, then sup{B(K)|K ∈KB}<∞.

Proof. Because A is continuous in K, by Lemma 1, and K ⊂K is compact, it follows that sup{A(K)|K ∈K}= max{A(K)|K ∈ K}<∞.

For any k ∈KB, let k∗ = max{k2, k3} and k∗ = min{k2, k3}. Then it follows from Theorem 2.1 of Laforgia

(1991) that

b(k1, k2, k3, y)<e−k1y√y

[e2(k∗−k∗)

√y k∗

k∗

]I1(k2

√y) = a(k1, k2,2(k∗− k∗), y).

Therefore B(k1, k2, k3)≤A(k1, k2,2(k∗− k∗)) for any k ∈KB. Moreover, the compactness of KB implies the

compactness of the set

K∗ := {(k1, k2,2(k∗− k∗))|(k1, k2, k3)∈ KB, k∗ = max{k2, k3}, k∗ = min{k2, k3}},

which is a subset of K. Therefore sup{B(K)|K ∈ KB} ≤ sup{A(K)|K ∈K∗}<∞.

Proposition 1. In the catastrophe bond example, if λ ≥ λ > 0 and 2θ > θ ≥ θ > 0, then for any target

state x∈R2,∫R2 σ

2x (x′)dπ(x′)<∞ and the sequence {σ2

x (Xn) , n= 1,2 · · · } is uniformly integrable.

Proof. Consider any target state x∈R2 and any sampling state x′ ∈R2. The likelihood ratio is `x(y;x′) =

h(y;x)/h(y;x′). Because the simulation output F (Yn) is between 0 and 1, for all n, the target-x-sample-x′

variance σ2x (x′) defined in Equation (5) satisfies

0≤ σ2x (x′)≤E

[(F (Yn) `x(Yn;x′))2|Xn = x′

]≤E

[(`x(Yn;x′))2|Xn = x′

].

To establish the desired conclusions, it suffices to show that this conditional second moment has a finite

upper bound over x′ ∈R2, for any fixed x∈R2. Denote (λ, θ) =ϕ(x)> 0 and (λ′, θ′) =ϕ(x′). We have

E[(`x(Yn;x′))2|Xn = x′

]=

(e−λ

e−λ′

)2

e−λ′+

∞∫

0

(h(y;x)

h(y;x′)

)2

h(y;x′)dy.

The first term is bounded above by eλ−2λ. For the second term, we have

∞∫0

(h(y;x)

h(y;x′)

)2

h(y;x′)dy =∞∫0

[√λθye−λ− y

θ I1

(2√

λyθ

)]2√

λ′θ′y e

−λ′− yθ′ I1

(2√λ′yθ′

)dy=√

λ2θ′

λ′θ2eλ′−2λ

∞∫0

1√ye−(

2θ′−θθθ′

)y

[I1

(2√

λyθ

)]2I1

(2√λ′yθ′

) dy

=√

λ2θ′

λ′θ2eλ′−2λB

(2θ′−θθθ′

,2√

λθ,2√

λ′

θ′

).

(16)

For all x′ ∈ R2, we have that (λ′, θ′) = ϕ(x′) is in a compact set [λ, λ]× [θ, θ] that does not contain zero.

On this set, the arguments of B are all bounded so{(

2θ′−θθθ′

,2√

λθ,2√

λ′

θ′

)|(λ′, θ′)∈ [λ, λ]× [θ, θ]

}= KB is

compact. Thus, it follows from the second claim of Lemma 2 that Equation (16) has a finite upper bound

over x′ ∈R2.


Proposition 2. In the catastrophe bond pricing example, if λ≥ λ> 0 and 2θ > θ≥ θ > 0, then Assump-

tions A1, A2, A3, and A4 in Theorem 4 are satisfied.

Proof. Since F (y) = 1{y≤K} + p1{y>K} ∈ [p,1] so µx = E [F (Yx)] ∈ [p,1] for all x ∈ X and therefore A1

holds.

Clearly, for y= 0, h(y;λ, θ) is bounded by e−λ and e−λ. For y ∈ (0,∞), the modified Bessel function of the

first kind I1(√λs/θ) is bounded over (λ, θ)∈ [λ, λ]× [θ, θ]. Therefore A2 holds.

Define k∗ = max{2√λ/θ|(λ, θ)∈ Ψ}, k∗ = max{2

√λ/θ|(λ, θ)∈ Ψ}. It follows from Theorem 2.1 in Laforgia

(1991) we have, for all λ,λ′ ∈ [λ, λ], θ, θ′ ∈ [θ, θ] and all y ∈ (0,∞),

ω(y) =

√λθye−λ−

yθ I1

(2√

λy

θ

)

√λ′

θ′ye−λ

′− yθ′ I1

(2√

λ′yθ′

) <

√λθ′

λ′θeλ′−λe(

1θ′−

1θ )y(k∗

k∗e(k∗−k∗)

√y

)≤ aeby+(k∗−k∗)

√y

where a= k∗

k∗

√λθλθeλ−λ and b= 1

θ− 1

θ. Let ε= 1

θ− 1

2θ> 0 and b= b+ ε= 1

2θ. Then we have ω(y)< aeby for

a= ae(k∗−k∗)2

4ε because (k∗− k∗)√s− εs≤ (k∗−k∗)2

4ε. Finally, ω(0) = eλ

′−λ <a. Therefore A3 holds.

For A4, consider any t < 1/θ and x ∈ X , the moment generating function for the compound CAT bond

loss is given by

M(x, t) = e−λ +∞∫0

ety√

λθye−λ−

yθ I1

(2√

λy

θ

)dy= e−λ

(1 +

√λθA(

1θ− t,2

√λθ,0))

. (17)

For all x∈R2, we have that (λ, θ) =ϕ(x) is in a compact set [λ, λ]× [θ, θ] that does not contain zero. On this

set, the arguments of A are all bounded, so{(

1θ− t,2

√λθ,0)|(λ, θ)∈ [λ, λ]× [θ, θ]

}= K is compact. Thus,

it follows from the first claim of Lemma 2 that Equation (17) has a finite upper bound over x∈R2.

Acknowledgments

Portions of this article were published in the Proceedings of the Winter Simulation Conference (Feng and

Staum 2015). The authors thank Shane Henderson and Barry Nelson for providing references. They are

grateful for financial support from the Intel Parallel Computing Center at Northwestern University.

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